Abstract

One of the several different notions of integrability for systems of PDEs is the so-called Liouville integrability, namely the existence of a suitable number of conserved quantities during the evolution of a system. Such a notion is analogue to the one of integrability for finite-dimensional systems, and it is bases on the Lie algebra defined by the Poisson bracket, which constitutes the Hamiltonian structure of the system.

I will review the notion of Poisson structures (in some contexts called Hamiltonian operators) focusing on systems of PDEs, showing their relation with the integrability of the system (bi-Hamiltonian pairs, recursion operators..) and underlying their common features.

One crucial feature of Poisson brackets for integrable PDEs is that they can be regarded as differential-geometrical objects on the space of functions between two manifolds and that their existence is constrained by the structure of the so-called “target manifold”. I will review the geometry of the Poisson brackets of hydrodynamic type defined by Dubrovin and Novikov and show how to naturally extend the notion to brackets defined by higher order and/or multidimensional differential operators. The case of weakly non-local Poisson structures of first and third order will be discussed too. Finally, I will present the first results of a joint project with G. Carlet about the structure of differential-geometric Poisson brackets of arbitrary order.

专家简介：Matteo Casati, Researcher at the School of Mathematics and Statistics, Ningbo University, PhD in Mathematical Physics from International School for Advanced Studies (SISSA) of Trieste, Italy (2015). Worked as a postdoctoral researcher at the Italian Institute of Higher Mathematics, Loughborough University and University of Kent (UK). His main research area are geometric and algebraic structure of integrable Hamiltonian systems; his research work have been published in several internationally renowned journals as Commun. Math. Phys., Nonlinearity, Stud. Appl. Math.